Symbolic template iterations of complex quadratic maps

Rǎdulescu, Anca; Pignatelli, Ariel

doi:10.1007/s11071-016-2626-3

Cited by 10 publications

(9 citation statements)

References 23 publications

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“…Due to extensive work spanning almost one century, from Julia [8] and Fatou [9] until recent developments [10,11], we now have the following: For a single iterated logistic map [12,13], the Fatou-Julia Theorem implies that the Julia set is either totally connected, for values of c in the Mandelbrot set (i.e., if the orbit of the critical point 0 is bounded), or totally disconnected, for values of c outside of the Mandelbrot set (i.e., if the orbit of the critical point 0 is unbounded). In previous work, the authors showed that this dichotomy breaks in the case of random iterations of two maps [19]. In our current work, we focus on extensions for networked logistic maps.…”

Section: Networking Logistic Mapsmentioning

confidence: 91%

Real and complex behavior for networks of coupled logistic maps

Rǎdulescu

Pignatelli

2016

Nonlinear Dyn

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Many natural systems are organized as networks, in which the nodes interact in a time-dependent fashion. The object of our study is to relate connectivity to the temporal behavior of a network in which the nodes are (real or complex) logistic maps, coupled according to a connectivity scheme that obeys certain constrains, but also incorporates random aspects. We investigate in particular the relationship between the system architecture and possible dynamics. In the current paper we focus on establishing the framework, terminology and pertinent questions for low-dimensional networks. A subsequent paper will further address the relationship between hardwiring and dynamics in high-dimensional networks.

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Section: Networking Logistic Mapsmentioning

confidence: 91%

Real and complex behavior for networks of coupled logistic maps

Rǎdulescu

Pignatelli

2016

Nonlinear Dyn

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“…It is worth noting that a great deal of work has been done on nonlinear dynamical system theory in the past decades. In this field, a series of problems are closely related to the Julia set, such as the size of the nonlinear attraction domain [27] and the boundary of iterative mapping [28]. Even if the system is very plain, some of the boundaries of the attractive domain may be extremely complicated [29].…”

Section: Introductionmentioning

confidence: 99%

Julia Sets and Their Control of Discrete Fractional SIRS Models

Ouyang

Zhang

2019

Complexity

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It is of crucial significance to study infectious disease phenomenon by using the discrete SIRS model with the Caputo deltas sense and fractal viewpoint. In this paper, Julia set of the discrete fractional SIRS model is established to analyze the fractal dynamics of this model. Then three different controllers, which are, respectively, added to different parts of the model as a whole, a part, and a product factor, are designed to change the Julia set, and the graphs illustrate the complexity of the model. Simulation results show the efficacy of these methods.

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“…As in our previous work [12], we consider here iterations of two different functions, f c 0 and f c 1 , according to a general binary symbolic sequence s (template), in which the "zero" positions correspond to iterating the function f c 0 and the "one" positions correspond to iterating the function f c 1 . We view template iterations as a more appropriate framework for replication or learning algorithms that appear in nature, with patterns that evolve in time, and which may involve occasional, random or periodic "errors."…”

Section: Introductionmentioning

confidence: 99%

“…One can study how the "sustainable set of features" (filled Julia set) changes when introducing occasional or periodic errors into the iteration process. For example, investigating the types of errors for which the Julia set is disconnected would help identify the systems capable of functioning in a sustainable range, when they are initiated within multiple connected loci of initial conditions [12].…”

Section: Introductionmentioning

confidence: 99%

“…In our previous work, we introduced the fixed template Mandelbrot set [12] as the locus in C 2 of (c 0 , c 1 ) for which the critical orbit is bounded under a fixed template s. We investigated some of the properties of this set and of its 2-dimensional complex slices for both periodic and random templates. For example, we suggested that there are detectable differences in the Hausdorff dimension along the boundary of Mandelbrot slices for fixed templates, between the case of periodic and that of non-periodic templates.…”

Section: Introductionmentioning

confidence: 99%

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Template iterations of quadratic maps and hybrid Mandelbrot sets

Rǎdulescu¹,

Kelsey²,

Williams³

2016

Preprint

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As a particular problem within the field of non-autonomous discrete systems, we consider iterations of two quadratic maps f c0 = z 2 + c 0 and f c1 = z 2 + c 1 , according to a prescribed binary sequence, which we call template. We study the asymptotic behavior of the critical orbits, and define the Mandelbrot set in this case as the locus for which these orbits are bounded. However, unlike in the case of single maps, this concept can be understood in several ways. For a fixed template, one may consider this locus as a subset of the parameter space in (c 0 , c 1 ) ∈ C 2 ; for fixed quadratic parameters, one may consider the set of templates which produce a bounded critical orbit. In this paper, we consider both situations, as well as hybrid combinations of them, we study basic topological properties of these sets and propose applications.Using random iterations of discrete functions, we build a mathematical framework that can be used to study the effect of errors in copying mechanisms (such as DNA replication). In our theoretical setup -in which one of the functions is the correct one, and the other one is the erroneous perturbation -we consider problems that a sustainable replication system may have to solve when facing the potential for errors. We find that it is possible to tell which specific errors are more likely to affect the system's dynamics, in absence of prior knowledge of their timing. Moreover, within an optimal locus for the correct function, almost no errors can affect the sustainability of the system. Mathematically, our work complements broader existing results in nonautonomous dynamics with more specific detail for the case of two random iterated functions, which is a valuable context for applications.

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Symbolic template iterations of complex quadratic maps

Cited by 10 publications

References 23 publications

Real and complex behavior for networks of coupled logistic maps

Real and complex behavior for networks of coupled logistic maps

Julia Sets and Their Control of Discrete Fractional SIRS Models

Template iterations of quadratic maps and hybrid Mandelbrot sets

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